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1. Continuum limit for the Ablowitz–Ladik system

   https://doi.org/10.1088/1361-6544/acd978  

   Killip, Rowan; Ouyang, Zhimeng; Visan, Monica; Wu, Lei (June 2023, Nonlinearity)

   Abstract We show that solutions to the Ablowitz–Ladik system converge to solutions of the cubic nonlinear Schrödinger equation for merely L 2 initial data. Furthermore, we consider initial data for this lattice model that excites Fourier modes near both critical points of the discrete dispersion relation and demonstrate convergence to a decoupled system of nonlinear Schrödinger equations. 

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2. On the Whitham system for the (2+1)‐dimensional nonlinear Schrödinger equation

   https://doi.org/10.1111/sapm.12543  

   Ablowitz, Mark J.; Cole, Justin T.; Rumanov, Igor (December 2022, Studies in Applied Mathematics)

   Abstract Whitham modulation equations are derived for the nonlinear Schrödinger equation in the plane ((2+1)‐dimensional nonlinear Schrödinger [2d NLS]) with small dispersion. The modulation equations are obtained in terms of both physical and Riemann‐type variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one‐dimensional traveling waves. In both the elliptic and hyperbolic cases, the traveling waves are found to be unstable. This result is consistent with previous investigations of stability by other methods and is supported by direct numerical calculations. 

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3. Stability of smooth solitary waves under intensity-dependent dispersion

   https://doi.org/10.1093/imamat/hxaf005  

   Kevrekidis, P G; Pelinovsky, D E; Ross, R M (December 2024, IMA Journal of Applied Mathematics)

   Abstract The nonlinear Schrödinger (NLS) equation in one dimension is considered in the presence of an intensity-dependent dispersion term. We study bright solitary waves with smooth profiles that extend from the limit where the dependence of the dispersion coefficient on the wave intensity is negligible to the limit where the solitary wave becomes singular due to vanishing dispersion coefficient. We analyse and numerically explore the stability for such smooth solitary waves, showing with the help of numerical approximations that the family of solitary waves becomes unstable in an intermediate region between the two limits, while being stable in both limits. This bistability, which has also been observed in other NLS equations with generalized nonlinearity, brings about interesting dynamical transitions from one stable branch to another stable branch, which are explored in direct numerical simulations of the NLS equation with the intensity-dependent dispersion term. 

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4. Integrable nonlocal derivative nonlinear Schrödinger equations

   https://doi.org/10.1088/1361-6420/ac5f75  

   Ablowitz, Mark J; Luo, Xu-Dan; Musslimani, Ziad H; Zhu, Yi (April 2022, Inverse Problems)

   Abstract Integrable standard and nonlocal derivative nonlinear Schrödinger equations are investigated. The direct and inverse scattering are constructed for these equations; included are both the Riemann–Hilbert and Gel’fand–Levitan–Marchenko approaches and soliton solutions. As a typical application, it is shown how these derivative NLS equations can be obtained as asymptotic limits from a nonlinear Klein–Gordon equation. 

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5. Vector modulation instability in birefringent graded-index multimode fibers

   https://doi.org/10.1364/JOSAB.412450  

   Ahsan, Amira_S; Agrawal, Govind_P (December 2020, Journal of the Optical Society of America B)

   We study vectorial modulation instability occurring inside a birefringent graded-index (GRIN) fiber when the two polarization components of the optical field are coupled nonlinearly through cross-phase modulation. In the scalar case in which only modes of one polarization are excited, the geometric parametric instability is known to produce an infinite number of sidebands around the wavelength of the input optical beam. We show that the birefringence of a GRIN fiber splits each of these sidebands into a triplet, whose frequency spacing depends on the differential group delay between the orthogonally polarized components. We verify the predictions of the linear stability analysis numerically by solving two coupled nonlinear Schrödinger equations that include spatial self-imaging effects through an effective nonlinear parameter. We present results for both continuous and pulsed optical beams experiencing normal or anomalous group-velocity dispersion inside a GRIN fiber. 

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